This week, the data set under scrutiny contains housing values in the suburbs of Boston in the late 1970s (more information about the data set can be found here). It is part of the MASS library in R. Let’s have a look at its variables:
## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
The data set consists of 506 neighbourhoods in Boston and for each there are 14 explanatory variables, such as crime rate per capita and nitric oxides concentration. Let’s then have a closer look at the variables and visualize them.
## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00
All of the variables are numerical. The chas variable is the only binary one, while the others have ranges of varying magnitudes. Some of the variable pairs are very strongly correlated, such as (rad, tax) and (nox, dis), while many pairs are relatively independent of each other.
In order to be able to do proper linear discriminant analysis (LDA) later, all the variables need to be scaled. This is done for each variable by subtracting the mean and dividing by the standard deviation. Here is how the variables look after scaling:
## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865
Now all the variables are of comparable magnitude. Next let’s take the crim variable, which is the per capita crime rate, and turn it into a categorical variable crime using the quantiles as break points. The four categories of approximately equal size are named low, med_low, med_high and high:
##
## low med_low med_high high
## 127 126 126 127
Before we attempt to classify the data with LDA, the data is divided into train and test sets with 80/20 split:
## Train set size: 404
## Test set size: 102
With the scaled training set, crime is used as a target variable for the LDA and all the other variables are predictor variables.
Let’s see how well our LDA model fares when evaluated on the test set:
test_y <- test$crime
test_x <- dplyr::select(test, -crime)
lda.pred <- predict(lda.fit, newdata = test_x)
table(correct = test_y, predicted = lda.pred$class)
## predicted
## correct low med_low med_high high
## low 23 10 1 0
## med_low 3 14 4 0
## med_high 0 11 12 1
## high 0 0 1 22
The table shows that the neighbourhoods with the highest crime rates are identified very reliably, while the neighbourhoods with lower crime rates cannot be classified as successfully. Most notably, the LDA cannot distinguish medium low and medium high crime rate areas convincingly based on these explanatory features, and one neighbourhood with low crime rate was even classified as a medium high crime rate area.
Let’s now for a moment forget about the LDA that was done and try to also cluster the original data by using the k-means clustering algorithm. The data is first scaled similary as earlier. Let’s have a look at the summary of the Euclidian distances in the scaled data set:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970
The minimum and maximum values indicate that some data points are indeed very close to each other when compared to the mean distance, but there’s also considerable distances between other points. This signifies that the data set is not homogeneous and the data points form some sorts of clusters. The optimal number of clusters can be investigated by running the k-means algorithm for a range of k and calculating the Within-Cluster-Sum-of-Squares (WCSS) value for each k. The plot for WCSS as a function k is shown below.
By design, the WCSS value decreases monotonously as k is increased. The optimal value for k can be found by considering the so-called elbow point, where there is a biggest drop in WCSS between two values. This implies that perhaps k = 2 is the optimal value, but k = 3 should work fine as well. After that, the slope becomes less steep at each point. The clusters can be visualized by looking at the variables and coloring the different groups. Here k = 3 was used:
The three categories certainly form their own clusters, though it is quite difficult to interpret how well it works because these plots are 2-dimensional, while the clusters exist in 14-dimensional space. Let us then combine LDA and the k-means clustering to see if LDA is capable of finding the clusters. All the 14 explanatory variables are used as predictors, while the thee cluster categories are used as a target.
From this plot we can see that as was observed earlier, there is clearly one cluster of data points separate from the majority of the data points. The rad and tax variables are the most influential separators for the clusters. It was seen in the earlier LDA biplot as well that rad was important in separating the high crime rate neighbourhoods from the others. We can verify whether cluster 1 and the high crime rate areas are indeed the same clusters by comparing two 3D plots of the LDA, one labeled with the crime categories and another with the k-means clusters. (You can rotate the plots by dragging them!)
It appears that that cluster 1 practically corresponds to the high crime rate neighbourhoods. The k-means clustering algorithm also did reasonable job with identifying the low crime rate neighbourhoods (cluster 2) and the medium low crime rate neighbourhoods (cluster 3). Of course the k-means algorithm operated with one category less than the LDA, so they cannot have a one-to-one correspondence.